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12 I ELEMENTS OF THE THEORY OF SETS

Suppose (Aj)_AeL is a partition of X; then it is clear that the relation

** there exists AeL such that x e A_A and y e A_A"
is an equivalence relation between x and y.

Conversely, let R be an equivalence relation in X, and let G cz X x X
be its graph (Section 1.3); for each x e X, the cross section G(x) (Section 1.3)
is called the class (or equivalence class) of x for R (or " mod R"). The set of
all subsets of X which can be written G(x) for some x e X is a subset of
«P(X) called the quotient set ofX by R and written X/R; the mapping x -» G(x)
is called the canonical (or natural) mapping of X into X/R; it is surjective
by definition. The family of subsets of X defined by the natural injection of
X/R into ^(X) is a partition of X, whose elements are the classes mod R.
Indeed, if z e G(x) n G(y), both relations R(x, z) and R(y, z) hold, hence
also R(z, y) (symmetry) and R(x, y) (transitivity), which proves that y e G(x);
this implies G(y) a G(x) (transitivity) and exchanging x and y one gets
G(x) c GOO, hence finally G(x)-G(y); as moreover x<=G(x) for every
:c e X (refiexivity), our assertion is proved.

For every mapping/of X into a set Y, the relation/(x) =f(x^f) is an
equivalence relation between x and x'.

Let (X_A)_AeL be a family of subsets of a set Y, and for each AeL, let
X_A = {A} x X_A (subset of L x Y); it is clear that the restriction to X^ of the
second projection pr₂ : L x Y -» Y is a bijection p^ of X_A on X_A. The subset
S = y X^ c L x Y is called the sum of the family (X_A) (not to be mistaken

AeL

for the union of that family ft; it is clear that (X^) is a partition of S. Usually,
X_A and X_A will be identified by the natural bijection /?_A. If, for every AeL,
W_A is a mapping of X^ into a set T, there is one and only one mapping u of
S into T which coincides with W_A in each X_A.

With the same notations, let us now consider the subset of the set Y^L(Section 1,4) consisting of all mappings A->x_A of L into Y such that, for
every AeL, one has X_A eX_A; this subset is called the product of the family
(X;)_A6L. and is written J] X_A; for each x = (X_A) e[] X_A, and every index

<*eL AeL

\L e L, one writes x^ = pr_M(x). More generally, for each nonempty subset
J of L, one writes prj(x) = (X_A)_A _e, (subfamily of x = (JC_A)_A _e _L). From the axiom
of choice (1.4.5) it follows that if X_A ^ 0 for every AeL, then f] X_A^ 0,

and each of the mappings prj is surjective. Furthermore, if J and L - J are
both nonempty, the mapping x->far₃(x)₉ pr_L_j(;c)) is a bijection ofJ^X^

AeL



	12 I ELEMENTS OF THE THEORY OF SETS Suppose (Aj)_AeL is a partition of X; then it is clear that the relation ** there exists AeL such that x e A_A and y e A_A" is an equivalence relation between x and y. Conversely, let R be an equivalence relation in X, and let G cz X x X be its graph (Section 1.3); for each x e X, the cross section G(x) (Section 1.3) is called the class (or equivalence class) of x for R (or " mod R"). The set of all subsets of X which can be written G(x) for some x e X is a subset of «P(X) called the quotient set ofX by R and written X/R; the mapping x -» G(x) is called the canonical (or natural) mapping of X into X/R; it is surjective by definition. The family of subsets of X defined by the natural injection of X/R into ^(X) is a partition of X, whose elements are the classes mod R. Indeed, if z e G(x) n G(y), both relations R(x, z) and R(y, z) hold, hence also R(z, y) (symmetry) and R(x, y) (transitivity), which proves that y e G(x); this implies G(y) a G(x) (transitivity) and exchanging x and y one gets G(x) c GOO, hence finally G(x)-G(y); as moreover x<=G(x) for every :c e X (refiexivity), our assertion is proved. For every mapping/of X into a set Y, the relation/(x) =f(x^f) is an equivalence relation between x and x'. Let (X_A)_AeL be a family of subsets of a set Y, and for each AeL, let X_A = {A} x X_A (subset of L x Y); it is clear that the restriction to X^ of the second projection pr₂ : L x Y -» Y is a bijection p^ of X_A on X_A. The subset S = y X^ c L x Y is called the sum of the family (X_A) (not to be mistaken AeL for the union of that family ft; it is clear that (X^) is a partition of S. Usually, X_A and X_A will be identified by the natural bijection /?_A. If, for every AeL, W_A is a mapping of X^ into a set T, there is one and only one mapping u of S into T which coincides with W_A in each X_A. With the same notations, let us now consider the subset of the set Y^L(Section 1,4) consisting of all mappings A->x_A of L into Y such that, for every AeL, one has X_A eX_A; this subset is called the product of the family (X;)_A6L. and is written J] X_A; for each x = (X_A) e[] X_A, and every index *<eL AeL** \L e L, one writes x^ = pr_M(x). More generally, for each nonempty subset J of L, one writes prj(x) = (X_A)_A _e, (subfamily of x = (JC_A)_A _e _L). From the axiom of choice (1.4.5) it follows that if X_A ^ 0 for every AeL, then f] X_A^ 0, and each of the mappings prj is surjective. Furthermore, if J and L - J are both nonempty, the mapping x->far₃(x)₉ pr_L_j(;c)) is a bijection ofJ^X^ AeL